Something in their story sounds fishy - if they loosened up the cam bolts, that would allow the valve springs to lift the cam up, closing the valves (assuming the cam lobe had them opened). Tightening the bolts back down would then potentially open the valves back up,allowing air to escape the cylinder, into either the intake or exhaust manifolds. A proper leak down test should be done with the cams removed, allowing all the valves to be closed - with the cams in place, there's always going to be at least one cylinder with a valve open somewhere.
On the lifters; yes, they do have a hydraulic element to them that has to pump up with oil - given that the average engine operates at approximately 20psi minimum oil pressure, at a flow rate of 1-2 gpm, all of the lifters should be pressurized and filled within a matter of seconds after the engine started- no way in hell it would take 15 minutes. And if anything, it would run smoother once they are pumped up.
Have they proven to you that when the crankshaft timing Mark is at TDC, the camshaft timing marks are properly aligned, and the tensioner position indicator is set correctly? If not, they are currently trying to force combustion byproduct into your rectum, in hopes you'll give up and give them lots of money.
This is all just my opinion, based on the information provided - but something about their latest attempt doesn't pass the smell test....